By applying the central limit theory to \(h(^*)\) , we can establish the asymptotic normality of \(\tilde{\theta }\)  in Theorems REF  and REF . When \(Q_n()\)  is the negative log-likelihood of the data, this approach has been successfully used in [1]} and, moreover, the estimator \(\tilde{\theta }\)  is asymptotically equivalent to the de-biased estimator in [2]}, [3]}. As will be explained in the following subsections, the analysis based on the log-likelihood becomes intractable for the latent graphical model and requires stringent technical conditions for the cluster-average graphical model. To overcome this difficulty, we employ the pseudo score functions relative to the empirical risk functions \(Q_n()\)  defined in (REF ) and (REF ). The resulting one-step estimator still attains the information bound established in the literature, and more importantly requires weaker technical assumptions than the existing methods. Moreover, in addition to (REF ), we also derive explicitly the speed at which the normal approximation is attained.
